4 Answers
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the word brienne screamed was “sword”; lady stoneheart gave brienne the choice of either swearing her sword to her or being hung, saying “sword or noose”, and as brienne was being hung she screamed “sword”

This can be inferred from the text, since as George said, Brienne was given the choice of two words in AFFC. And we

See her later on in ADWD, implying she said the word Lady Stoneheart wanted to hear.

She shouts 'sword'. She was given a choice of being Lady Stoneheart's 'sword' and going to get Jamie for her, or of dying by the 'noose'. She shouts sword which is what saves her - and saves Podrick. George explained this to an audience at a recent chat.

the word is a name - Arya. Catelyn thinks Arya dead and so did most everyone else. The Elder Brother on the Silent Isle shocked Brienne with his news. Brienne was in bad shape and shocked by Lady Catelyn's appearance and that forestalled her sharing that information but the prospect of imminent death tends to sharpen the senses.